Nonexistence of Permutation Binomials of Certain Shapes

Ariane M. Masuda, Michael E. Zieve

Abstract

Suppose $x^m+ax^n$ is a permutation polynomial over ${\Bbb F}_p$, where $p>5$ is prime and $m>n>0$ and $a\in{\Bbb F}_p^*$. We prove that $\gcd(m-n,p-1)\notin\{2,4\}$. In the special case that either $(p-1)/2$ or $(p-1)/4$ is prime, this was conjectured in a recent paper by Masuda, Panario and Wang.